Journal of Advanced Research (2015) 6, 895–905
Cairo University
Journal of Advanced Research
ORIGINAL ARTICLE
Exponentiated power Lindley distribution Samir K. Ashour a b
a,*
, Mahmoud A. Eltehiwy
b
Institute of Statistical Studies & Research, Department of Mathematical Statistics, Cairo University, Egypt Faculty of Commerce, Department of Statistics, South Valley University, Egypt
A R T I C L E
I N F O
Article history: Received 15 July 2014 Received in revised form 11 August 2014 Accepted 17 August 2014 Available online 24 August 2014 Keywords: Lambert function Least square estimation Maximum likelihood estimation Order statistics Power Lindley distribution
A B S T R A C T A new generalization of the Lindley distribution is recently proposed by Ghitany et al. [1], called as the power Lindley distribution. Another generalization of the Lindley distribution was introduced by Nadarajah et al. [2], named as the generalized Lindley distribution. This paper proposes a more generalization of the Lindley distribution which generalizes the two. We refer to this new generalization as the exponentiated power Lindley distribution. The new distribution is important since it contains as special sub-models some widely well-known distributions in addition to the above two models, such as the Lindley distribution among many others. It also provides more flexibility to analyze complex real data sets. We study some statistical properties for the new distribution. We discuss maximum likelihood estimation of the distribution parameters. Least square estimation is used to evaluate the parameters. Three algorithms are proposed for generating random data from the proposed distribution. An application of the model to a real data set is analyzed using the new distribution, which shows that the exponentiated power Lindley distribution can be used quite effectively in analyzing real lifetime data. ª 2014 Production and hosting by Elsevier B.V. on behalf of Cairo University.
Abbreviations: EPLD, Exponentiated power Lindley distribution; PLD, Power Lindley distribution; BGLD, Beta generalized Lindley distribution; GLD, Generalized Lindley distribution; LD, Lindley distribution; MW, Modified Weibull distribution; WD, Weibull distribution; EE, Exponentiated exponential distribution; Cdf, Cumulative distribution function; pdf, Probability density function; h(t), Hazard rate function; Q(p), Quantile function; E(Xr), The rth moment; MX(t), The moment generating function; MLE, Maximum likelihood estimator; LSE, Least square estimator; MSE, mean square error; L, Log-likelihood function; K–S, Kolmogorov– Smirnov test; AIC, Akaike information criterion; BIC, Bayesian information criterion. * Corresponding author. E-mail address: [email protected] (S.K. Ashour). Peer review under responsibility of Cairo University.
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Introduction Lindley [3], introduced a one-parameter distribution, known as Lindley distribution, given by its probability density function gðt; hÞ ¼
h2 ð1 þ tÞeht ; 1þh
t > 0;
h > 0:
ð1Þ
It can be seen that this distribution is a mixture of exponential (h) and gamma (2, h) distributions. Its cumulative distribution function has been obtained as GðtÞ ¼ 1
h þ 1 þ ht ht e ; hþ1
t > 0;
h > 0:
ð2Þ
Ghitany et al. [4] have discussed various properties of this distribution and showed that in many ways that the pdf given by (1) provides a better model for some applications than the
2090-1232 ª 2014 Production and hosting by Elsevier B.V. on behalf of Cairo University. http://dx.doi.org/10.1016/j.jare.2014.08.005
896
S.K. Ashour and M.A. Eltehiwy
exponential distribution. Bakouch et al. [5] obtained an extended Lindley distribution and discussed its various properties and applications. Ghitany et al. [6] developed a two-parameter weighted Lindley distribution and discussed its applications to survival data. Nadarajah et al. [2] obtained a generalized Lindley distribution and discussed its various properties and applications. Merovci and Elbatal [7] use the quadratic rank transmutation map in order to generate a flexible family of probability distributions taking Lindley-geometric distribution as the base value distribution by introducing a new parameter that would offer more distributional flexibility and called it transmuted Lindley-geometric distribution. Asgharzadeh et al. [8] introduced a general family of continuous lifetime distributions by compounding any continuous distribution and the Poisson–Lindley distribution. Oluyede and Yang [9] proposed a new four-parameter class of generalized Lindley (GLD) distribution called the beta-generalized Lindley distribution (BGLD). This class of distributions contains the betaLindley, GLD and Lindley distributions as special cases. A two parameter power Lindley distribution (PLD), of which the Lindley distribution ‘Eq. (1)’ is a particular case, has been suggested by Ghitany et al. [1]. They introduced a new extension of the Lindley distribution by considering the power transformation of the r.v. Y = T1/b. The pdf of the Y is readily obtained to be power Lindley distribution with parameters b and h and is defined by its probability density function pdf gðy; hÞ ¼
h2 bð1 þ yb Þ b1 hyb y e ; hþ1
y > 0;
h;
b > 0;
ð3Þ
It can easily be seen that at b = 1 , Eq. (3) reduces to the Lindley distribution. From Eq. (2), we see that the power Lindley distribution is a two-component mixture of Weibull distribution (with shape b and scale h), and a generalized gamma distribution (with shape parameters 2 b and scale h), with mixing h proportion p ¼ 1þh . gðy; a; hÞ ¼ pf1 ðyÞ þ ð1 pÞf2 ðyÞ; where p ¼
ð4Þ
h , 1þh b
f1 ðyÞ ¼ hbyb1 ehy ;
y > 0;
h;
b > 0;
ð5Þ
and b
f2 ðyÞ ¼ h2 by2b1 ehy ;
y > 0;
h;
b > 0:
ð6Þ
In this paper, we introduce a new distribution with three parameters, referred to as the exponentiated power Lindley (EPLD) distribution, with the hope that it will attract many applications in different disciplines such as survival analysis, reliability, biology and others. One of the main goals to introduce this new distribution was that it involves three distributions as sub-models. Generally, the EPLD distribution generalizes the generalized Lindley (GLD) [2], power Lindley (PLD) [1] and Lindley (LD) [3] distributions. The third parameter indexed to this distribution makes it more flexible to describe different types of real data than its sub-models. The EPLD distribution, due to its flexibility in accommodating different forms of the hazard function, seems to be a suitable distribution that can be used in a variety of problems in fitting survival data. The EPLD distribution is not only convenient for fitting comfortable bathtub-shaped failure rate data but
also suitable for testing goodness-of-fit of some special submodels such as the PLD, GLD and LD distributions. The new extension of the power Lindley distribution is most conveniently specified in terms of the cumulative distribution function: a hxb b ehx ; FðxÞ ¼ 1 1 þ ð7Þ hþ1 for x > 0, h, b, a > 0 and the corresponding probability density function (pdf) is given by a1 ah2 bxb1 hxb b b 1 þ xb ehx 1 1 þ ehx fðxÞ ¼ ; ð8Þ hþ1 hþ1 The corresponding hazard rate function is ah2 b bxb1 1 þ xb ehx ðh þ 1Þ a1 hxb b 1 ehx 1 1þ fSðxÞg ; hþ1
hðxÞ ¼
ð9Þ
where a hxb b ehx : SðxÞ ¼ 1 1 1 þ hþ1
ð10Þ
Note that Eq. (8) has closed form survival functions and hazard rate functions. For (b = 1) , (a = 1) and (a = b = 1) we have the pdfs of generalized Lindley distribution , power Lindley and Lindley distributions respectively. As we shall see later, Eq. (8) has the attractive feature of allowing for monotonically decreasing; monotonically increasing and bath tub shaped hazard rate functions while not allowing for constant hazard rate functions. Another motivation for the new distribution in (Eq. (7)) can be described as follows. Consider the two parameters power Lindley distribution [1] specified by the cumulative distribution function: hxb b ehx ; ð11Þ FpL ðxÞ ¼ 1 1 þ hþ1 for x > 0 and h, b > 0. Suppose X1, X2 , . . . , Xa are independent random variables distributed according to (Eq. (11)) and represent the failure times of the components of a series system, assumed to be independent. Then the probability that the system will fail before time x is given by PrðmaxðX1 ;X2 ;. .. ;Xa Þ 6 xÞ ¼ PrðX1 6 xÞPrðX2 6 xÞ . .. PrðXa 6 xÞ ¼ FpL ðxÞFpL ðxÞ a hxb b . .. FpL ðxÞ ¼ 1 1 þ ehx : hþ1
So, Eq. (7) gives the distribution of the failure of a series system with independent components. From Eqs. (4)–(6), it follows that cumulative distribution function, Eq. (7), can be represented as 1 X a ai ai i FðxÞ ¼ ðxÞFGG2 ðxÞ; p ð1 pÞi FW i i¼0 b
where FW ðxÞ ¼ 1 eðhx Þ is the cumulative distribution function of Weibull random variable with shape b and scale h
Exponentiated power Lindley distribution
897
b
w(y) = [F(y)]a1 . The weight function is an increasing or decreasing if a > 1 or a < 1 respectively. Therefore, if Y is decreasing pdf then for a > 1, The pdf of X also is a decreasing pdf. If Y has a unimodal pdf, then Mode(X) > ( ( ( 0; a 6 1 ; II. 12 < b < 1; h P g1 ðbÞ; a 6 1 , pffiffiffiffiffiffiffiffiffiffi 12 bð1bÞ , where g1 ðbÞ ¼ b III. {b = 1, h P 1, a = 1}; IV. {b P 1, h > 0, a < 1}. b. Unimodal if I. {b = 1, h > 0, a > 1}; II. {b = 1, 0 < h < 1, a = 1}; III. {b > 1, h > 0, a P 1}. c. Decreasing–increasing–decreasing if 12 < b < 1; 0 < h < g1 ðbÞ; a ¼ 1Þ. Fig. 1 illustrates some of the possible shapes of the density function of exponentiated power Lindley distribution for selected values of the parameters (h, b, a). The behavior of h(x) at x = 0 and x = 1, respectively, are given by
Shapes In this section, we discuss the shape characteristics of the pdf f(x) in Eq. (8) of the EPL(h, b, a) distribution. The behavior of f(x) at x = 0 and x = 1, respectively, are given by 8 > < 1; if b < 1 or a < 1; h2 fð0Þ ¼ hþ1 ; if b ¼ 1 and a ¼ 1 ; fð1Þ ¼ 0: > : 0; if b > 1 or a > 1;
hð0Þ ¼
8 > < 1;
if b < 1 or a 6 1;
h2 ; > hþ1
if b ¼ 1 and a ¼ 1; : 0; if b > 1 or a P 1; 8 if b < 1 or a 6 1; > < 0; h2 ; if b ¼ 1 and a ¼ 1; hð1Þ ¼ hþ1 > : 1; if b > 1 or a P 1;
If X has the pdf (8), the pdf of X is a weighted version of the pdf of Y in Eq. (3) and the weight function in this case is 1.5
f ( x , 1 , 0.2 , .9) f ( x , 2 , 1 , 1) f ( x , 3 , 2 , .5)
0.8
f ( x , 1 , 1 , 2)
1
0.6
f ( x , .6 , 1 , 1)
f ( x , 1.5 , 2 , 1.5) 0.4
0.5
0.2 0
0
1
2
x
3
0
4
0
2
4
x
6
8
0.12 f ( x , .3 , .85 , 1) f ( x , .35 , .9 , 1)
0.1 0.08 0.06 0
Fig. 1
2
4
x
6
8
Probability density function of the EPLD(h, b, a) for selected values of the parameters.
898
S.K. Ashour and M.A. Eltehiwy 2
4
1.5
h ( x , 1 , 1 , 1.5) h ( x , .1 , 2 , 1)
3
h ( x , 2 , .6 , 1)
1
h ( x , 10 , .2 , .8)
0.5 0
2 1
0
2
4
x
6
8
0
10
0
2
0
1
4
6
8
2
3
4
x
10 8
1
h ( x , 1.2 , .88 , 1) h ( x , 1.5 , .9 , 1)
h ( x , 2 , 2 , .05) 6
0.9
h ( x , 1 , 3 , .09) 4
0.8 0.7 0
Fig. 2
2 2
4
x
6
8
0
10
x
Hazard rate function of the EPLD(h, b, a) for selected values of the parameters.
Ghitany et al .[1] discussed and proved the cases in which the hazard rate function of the power Lindley distribution is decreasing, increasing and decreasing–increasing–decreasing. It follows that the HRF h(x) in Eq. (9) of the EPLD distribution is (a) Decreasing if I. 0 < b 6 12 ; h > 0; a 6 1 ; ð2b1Þ2 II. 12 < b < 1; h P g2 ðbÞ; a 6 1 , where g2 ðbÞ ¼ 4bð1bÞ ; (b) Increasing if {b P 1, h > 0, a P 1}. (c) Decreasing–increasing–decreasing if 12 < b < 1; 0 < h < g2 ðbÞ; a ¼ 1g. Fig. 2 shows the HRF h(x) of the exponentiated power Lindley distribution for some choices of values of the parameters (h, b, a). Stochastic orders
Theorem 1. Let X1 EPLD(h1, b, a1) and X2 EPLD (h2, b, a2). If h1 = h2, and a1 > a2 (or if a1 = a2 and h1 P h2), then X1 6 LrX2 and hence X1 6 hrX2 and X1 6 stX2. Proof f1 ðxÞ a1 h21 ðh2 þ 1Þb1 xb1 1 b b ¼ ð1 þ xb1 Þeðh2 x 2 h1 x 1 Þ f2 ðxÞ a2 h22 ðh1 þ 1Þb2 xb2 1 a1 1 h1 xb1 b eh1 x 1 1 1þ ðh1 þ 1Þ 1a2 h2 xb2 b eh2 x 2 1 1þ ; ðh2 þ 1Þ thus @ f ðxÞ ¼ ln 1 @x f2 ðxÞ
Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behavior. Suppose Xi is distributed according to Eqs. (7) and (8) with common shape parameter b and parameters hi and ai for i = 1, 2. Let Fi denote the cumulative distribution of Xi and let fi denote the probability density function of Xi. A random variable X1 is said to be smaller than a random variable X2 in the I. Stochastic order ðX 1 6st X 2 Þ if F1(x) P F2(x) for all x. II. Hazard rate order ðX 1 6hr X 2 Þ if h1(x) P h2(x) for all x. III. Likelihood ratio order ðX 1 6Lr X 2 Þ if ff 12 ðxÞ decreases in x. ðxÞ The following results due to Shaked and Shanthikumar [10] are well known for establishing stochastic ordering of distributions X1 6Lr X2 ) X1 6hr X2 ) X1 6st X2 : The EPLD is ordered with respect to the strongest ‘‘likelihood ratio’’ ordering as shown in the following theorem:
b1 h1 xb1 1 b2 h2 xb2 1 b1 b2 þ h1 xb1 h2 xb2 x b þb 1 b1 1 1 2 x ðb1 b2 Þ þ b1 x b2 xb2 1 þ ðxb1 þ 1Þðxb2 þ 1Þ b
ða1 1Þh21 b1 xb1 1 ð1 þ xb1 Þeh1 x 1 b b ðh1 þ 1Þ 1 1 þ hh11xþ11 eh1 x 1
þ
ða2 1Þh22 b2 xb2 1 ð1 þ xb2 Þeh2 x 2 : b b ðh2 þ 1Þ 1 1 þ hh22xþ12 eh2 x 2
b
@ Case (i) if h1 = h2, and a1 > a2, then @x ln ff12 ðxÞ < 0. This means ðxÞ
that X1 6Lr X2 and hence X1 6hr X2 and X1 6st X2. @ Case (ii) if h1 P h2, and a1 = a2 = a, then @x ln ff1 ðxÞ ðxÞ < 0. 2
This means that X1 6Lr X2 and hence X1 6hr X2 and X1 6st X2.
h
Moments Theorem 2. The rth moments E(Xr) of a exponentiated power Lindley random variable X, is given by
Exponentiated power Lindley distribution
899
a EðXr Þ ¼ EðXr Þ ¼ Ci;k hþ1 2 3 C rþbðkþ1Þ C rþbðkþ2Þ b b 4 5; þ rþkb rþbðkþ1Þ rþbðkþ2Þ rþkb hð b 1Þ :ði þ 1ÞCð b Þ hð b Þ :ði þ 1ÞCð b Þ where Ci;k ¼
P1 P1 i¼0
k¼0
a1 i
Theorem 3. Let X have an exponentiated power Lindley distribution. Then the moment generating function of X, MX(t), is 3 2 C jþbðkþ1Þ C jþbðkþ2Þ b b a 7 6 MX ðtÞ ¼ Ci;k;j 4 jþkb þ jþkb 5; jþbðkþ1Þ jþbðkþ2Þ hþ1 ði þ 1ÞCð b Þ h b 1 ði þ 1ÞCð b Þ hb
k i h ð1Þi hþ1 k
Proof Z
1
xr fðxÞdx;
MX ðtÞ ¼
0
k¼0
j¼0
k i h tj ð1Þi hþ1 . j! k
a1 i
Z
Z
1
etx fðxÞdx; 0
a1 ah2 bxrþb1 hxb b b 1 þ xb ehx 1 1 þ ehx dx hþ1 hþ1 0 Z 1 2 rþb1 a1 ah bx hxb b b ¼ ehx ehx 1 1 þ dx hþ1 hþ1 0 a1 Z 1 2 rþ2b1 ah bx hxb b b þ ehx ehx 1 1 þ dx: ð12Þ hþ1 hþ1 0
EðXr Þ ¼
1
MX ðtÞ ¼
Z
1 0
¼
Z
1 0
þ
Z
a1 ah2 bxb1 hxb b b 1 þ xb etx ehx 1 1 þ ehx dx hþ1 hþ1 a1 ah2 bxb1 tx hxb hxb b ehx e e 1 1þ dx hþ1 hþ1
1
0
Using the following binomial h ia1 b hxb ehx given by 1 1 þ hþ1
series
expansion
of
a1 X i 1 a1 hxb hxb b b 1 1þ ¼ eihx ; ehx ð1Þi 1 þ hþ1 h þ 1 i i¼0
Eq. (12) takes the following form k Z 1 2 b1þrþkb 1 X i h ah bx b ehx ðiþ1Þ dx h þ 1 h þ 1 i k 0 i¼0 k¼0 k Z 1 2 2b1þrþkb 1 1 X X a1 i h ah bx b ð1Þi ehx ðiþ1Þ dx þ hþ1 hþ1 i 0 i¼0 k¼0 k
EðXr Þ ¼
1 X a1
Let t = x and Ci;k ¼
P1
i¼0
k P a1 i h ð1Þi 1 . k¼0 k hþ1 i
EðX Þ ¼ Ci;k
Z
1
0
rþkb
ah2 t b htðiþ1Þ e dt þ Ci;k hþ1
Z 0
1
of
Eq. (13) takes the following form MX ðtÞ ¼
1 a1 X
þ
! i
ð1Þ
i
i¼0
1 a1 X
1 X
i
k¼0
k
! ð1Þi
i
i¼0
1 X k¼0
!
h hþ1
! i k
k Z
1 0
h hþ1
k Z
1 0
ah2 bxb1þkb tx hxb ðiþ1Þ e e dx hþ1 ah2 bx2b1þkb tx hxb ðiþ1Þ e e dx: hþ1
Using the following expansion of etx given by 1 j j X tx j¼0
j!
:
Eq. (13) can be rewritten as follows:
rþbðkþ1Þ
ah2 t b hþ1
expansion
h ii hxb and using the following binomial series expansion of 1 þ hþ1 given by i X k 1 i hxb h 1þ ¼ xkb : hþ1 hþ1 k k¼0
etx ¼
Eq. (12) can be rewritten as follows:
series
a1 X i 1 a1 hxb hxb b b i ehx ð1Þ 1 þ 1 1þ ¼ eihx ; hþ1 hþ1 i i¼0
ð1Þi
b
a1 ah2 bx2b1 tx hxb hxb b ehx e e 1 1þ dx: ð13Þ hþ1 hþ1
Using the following binomial h ia1 b hxb ehx given by 1 1 þ hþ1
h ii hxb and using the following binomial series expansion of 1 þ hþ1 given by i X k 1 i hxb h 1þ ¼ xkb ; h þ 1 hþ1 k k¼0
r
i¼0
Proof
EðXr Þ ¼
P1 P1 P1
where Ci;k;j ¼
htðiþ1Þ
e
dt; MX ðtÞ ¼
1 a1 X
3 C rþbðkþ1Þ C rþbðkþ2Þ b b a 6 7 EðXr Þ ¼ Ci;k 4 rþkb þ rþkb 5: rþbðkþ1Þ rþbðkþ2Þ hþ1 h b 1 ði þ 1ÞCð b Þ h b ; ði þ 1ÞCð b Þ
þ
1 X
a1
b
1 X
i
k¼0
k
!
i
i¼0
Let t = x k h tj . j! hþ1
i
ð1Þ
i
i¼0
C rþbðkþ1Þ b a a r EðX Þ ¼ Ci;k rþkb þ Ci;k rþbðkþ1Þ Cð b Þ 1Þ hþ1 h þ 1 ð b ði þ 1Þ h C rþbðkþ2Þ b rþkb ; rþbðkþ2Þ ð Þ b h ði þ 1ÞCð b Þ
!
ð1Þi
!
1 X
i
k¼0
k
h hþ1
!
and Ci;k;j ¼
k X 1 j Z 1 t ah2 bxb1þkbþj hxb ðiþ1Þ e dx j! hþ1 0 j¼0
h hþ1
k X 1 j Z 1 t ah2 bx2b1þkbþj hxb ðiþ1Þ e dx: j! hþ1 0 j¼0
P1 P1 P1 i¼0
k¼0
j¼0
a1 i
i ð1Þi k
Eq. (13) can be rewritten as follows:
2
MX ðtÞ ¼ Ci;k;j
Z
1 0
kbþj
ah2 t b htðiþ1Þ e dx þ Ci;k hþ1
Z
1 0
bðkþ1Þþj
ah2 t b hþ1
ehtðiþ1Þ dx;
900
S.K. Ashour and M.A. Eltehiwy
jþbðkþ1Þ
Algorithm I. (mixture form of the Lindley distribution)
C b a MX ðtÞ ¼ Ci;k;j jþkb jþbðkþ1Þ 1 hþ1 ði þ 1ÞCð b Þ hb
þ
jþbðkþ2Þ b
1. 2. 3. 4.
C a Ci;k;j jþkb ; jþbðkþ2Þ hþ1 ði þ 1ÞCð b Þ hb
2 3 C jþbðkþ1Þ C jþbðkþ2Þ b b a 6 7 MX ðtÞ ¼ Ci;k;j 4 jþkb þ jþkb 5: jþbðkþ1Þ jþbðkþ2Þ hþ1 ði þ 1ÞCð b Þ h b 1 ði þ 1ÞCð b Þ hb
Quantile function Let X denotes a random variable with the probability density function (Eq. (8)). The quantile function, say Q(p), defined by F(Q(p)) = p is the root of the equation # n o h½QðpÞb exp h½QðpÞb ¼ 1 p1=a ; 1þ 1þh
Generate Ui Unifrom(0, 1), i = 1, . . . , n; Generate Vi Exponential(h), i = 1, . . . n; Generate Gi Gamma(2, h), i = 1, . . . n; 1=a 1=b h , then set Xi ¼ Vi , otherwise, If Ui 6 p ¼ 1þh 1=b set Xi ¼ Gi ; i ¼ 1; . . . n.
Algorithm II. (mixture form of the power Lindley distribution) 1. 2. 3. 4.
Generate Ui Unifrom(0, 1), i = 1, . . . , n; Generate Yi Weibul(h, b), i = 1, . . . n; Generate Si GG(2, h, b), i = 1, . . . n; 1=a h If Ui 6 p ¼ 1þh , then set Xi = Yi, otherwise, set Xi = Si,i = 1, . . . n.
"
ð14Þ
for 0 < p < 1. Substituting Z(p) = 1 h h[Q(p)]b, one can rewrite Eq. (14) as
Algorithm III. (inverse cdf) 1. Generate Ui Unifrom(0, 1), i = 1, . . . , n; 2. Set
ZðpÞexpfZðpÞg ¼ ð1 þ hÞexpð1 hÞð1 p1=a Þ;
Xi ¼
for 0 < p < 1. So, the solution for Z(p) is ZðpÞ ¼ Wðð1 þ hÞexpð1 hÞð1 p1=a ÞÞ;
i1=b 1 1 h 1 W ð1 þ hÞexpð1 hÞ 1 U1=a ; i h h
ð15Þ where W(.) denotes the Lambert W function.
for 0 < p < 1, where W(.) is the Lambert W function, see Corless et al. [11] for detailed properties. Inverting Eq. (15), one obtains 1=b 1 1 1=a QðpÞ ¼ 1 Wðð1 þ hÞexpð1 hÞð1 p ÞÞ ; h h ð16Þ for 0 < p < 1. The particular case of Eq. (16) for (a = b = 1) has been derived recently by Jodra´ [12].
Algorithm IV 1. Generate Ui Unifrom(0, 1), i = 1, . . . , n; 2. Solve numerically the following equation in v 2 (0, 1):
vi ½h þ 1 ln vi ¼ 0: ðh þ 1Þ 1 U1=a i 3. Set Xi ¼
Generation algorithms Here, we consider simulating values of a random variable X with the probability density function in Eq. (8). Let U denote a uniform random variable on the interval (0, 1). One way to simulate values of X is to set hxb exp hxb ¼ 1 U1=a ; 1þ 1þh and solve for X, i.e. use the inversion method. Using Eq. (16), we obtain X as X¼
1=b 1 1 1 W ð1 þ hÞexpð1 hÞ 1 U1=a ; h h
where W[.] denotes the Lambert W function. We propose three different algorithms for generating random data from the exponentiated power Lindley distribution:
ln vi 1=b h
.
Maximum likelihood estimation of parameters Let x1, . . . , xn be a random sample of size n from EPLD. Then, the log-likelihood function is given by
Lða; b; hÞ ¼
n X
ln fðxi Þ;
i¼1
¼ n½ln ðaÞ þ ln ðbÞ þ 2 lnðhÞ lnðh þ 1Þ n n X X ln 1 þ xbi þ ðb 1Þ ln ðxi Þ þ i¼1
i¼1
n n X X ln Ai ðb; hÞ: h xbi þ ða 1Þ i¼1
i¼1
h i b hxb where Ai ðb; hÞ ¼ 1 1 þ hþ1i ehxi ; i ¼ 1; . . . ; n.
ð17Þ
Exponentiated power Lindley distribution
901
^ ^ The MLEs ^h; b; a of h, b, a are then the solutions of the following non-linear equations: n @ nðh þ 2Þ X Lða;b;hÞ ¼ xb @h hðh þ 1Þ i¼1 i " # n ða 1Þ X Ai;h ðb;hÞ þ ¼ 0; ð1 þ hÞ2 i¼1 Ai ðb;hÞ
ð18Þ
n n @ n X xbi ln ðxi Þ X Lða; b; hÞ ¼ þ þ ln ðxi Þ b @b b i¼1 x þ 1 i¼1 n n X X Ai;b ðb; hÞ ¼ 0; h xbi :ln ðxi Þ þ ða 1Þ Ai ðb; hÞ i¼1 i¼1
ð19Þ
n @ n X Lða; b; hÞ ¼ þ ln Ai ðb; hÞ ¼ 0; @a a i¼1
ð20Þ
where
2 n n n X @2L X xbi ln ðxi Þ2 n X x2b 2 b i ln ðxi Þ ¼ hx ln ð x Þ i i 2 @b2 b2 i¼1 xbi þ 1 xbi þ 1 i¼1 i¼1 h i
2 n ða 1Þ Ai ðb; hÞ:A 2 ðb; hÞ Ai;b ðb; hÞ X i;b þ ; ½Ai ðb; hÞ2 i¼1
" # @2L 2 1 ¼ n 2 @h2 h ðh þ 1Þ2 h i 2 n ða 1Þ Ai ðb; hÞA 2 ðb; hÞ ½Ai;h ðb; hÞ X i;h ; þ ½Ai ðb; hÞ2 i¼1
n X @2L Ai;h;a ðb; hÞ ¼ ; @h@a Ai ðb; hÞ i¼1
From (20) we can obtain the MLE of a as a function of (b, h), say ^aðb; hÞ, where n : ln Ai ðb; hÞ i¼1
@ 2 L n ¼ 2 ; @a2 a
n X @2L Ai;b;a ðb; hÞ ¼ ; @b@a Ai ðb; hÞ i¼1
b @Ai ðb; hÞ Ai;h ðb; hÞ ¼ ¼ ehxi hð1 þ hÞ 1 þ h þ hxbi xbi ; @h " # h xbi þ 1 @Ai ðb; hÞ b hxbi : Ai;b ðb; hÞ ¼ ¼ hxi e ln ðxi Þ @b hþ1
^ aðb; hÞ ¼ Pn
The second derivatives of L can be derived as follows:
ð21Þ
Putting ^aðb; hÞ in (17), we obtain n X gðb; hÞ ¼ L½^aðb; hÞ; b; h ¼ C n ln ½ ln Ai ðb;hÞ
n X @2L ¼ xbi ln ðxi Þ @h@b i¼1
n X ða 1Þ Ai ðb; hÞAi;h;b ðb; hÞ Ai;h ðb; hÞAi;b ðb; hÞ þ ; ½Ai ðb; hÞ2 i¼1
where
i¼1
n X
þ
b
Ai;h2 ðb; hÞ ¼
ln Ai ðb; hÞ þ n½ln ðbÞ þ 2 lnðhÞ lnðh þ 1Þ
i¼1 n X
n n X X ln 1 þ xbi þ ðb 1Þ ln ðxi Þ h xbi :
i¼1
i¼1
ð22Þ
i¼1
^MLE ; ^hMLE , can be obtained Therefore, the MLE of b, h, say b by maximizing (22) with respect to b and h. For the three parameters exponentiated power Lindley distribution EPLD(h, b, a), all the second order derivatives exist. Thus we have the inverse dispersion matrix is 20 1 0 13 0 1 ^ b11 V b12 V b13 h h V 7 6B C B b B^C b22 V b23 C A5; @ b A N4@ b A; @ V V 21 a
^ a 20
V1
V11 6B ¼ E4@ . . . V31
b31 V
... ... ...
b32 V
13 0 @2 L V13 2 C7 B @h . . . A5 ¼ E@ . . . @2 L V33 @h@a
...
@2 L @h@a
@2L @h@b @2L V23 ¼ @b@a V12 ¼
@2L ; @h@a @2L V33 ¼ 2 : @a V13 ¼
b hxi þ x2b i e
ðh þ 1Þ3
b Ai;b2 ðb; hÞ ¼ hxbi ln ðxi Þ2 ehxi
("
" b
Ai;b;a ðb; hÞ ¼ hxbi ln ðxi Þehxi " Ai;h;a ðb; hÞ ¼
... ...
@2 L @a2
1
"
2
!
ðh þ 1Þ2
!# hxbi 1þ ; hþ1
! # " #) h2 x2b 2hxbi 1 b i þ hxi 1 ; hþ1 hþ1
# hxbi þ h ; hþ1
! # hxbi 1 ; þ1 hþ1 ðh þ 1Þ2
b xbi ehxi
b
C . . . A:
Eq. (23) is the variance covariance matrix of the EPLD(h, b, a) @2L @h2 @2L V22 ¼ 2 @b
!
b33 V
ð23Þ
V11 ¼
2xbi ehxi
"
Ai;h;b ðb;hÞ ¼ xbi ehxi ln ðxi Þ
! !# hxbi hxb 1 hxbi þ h : þ 1 þ i 2 hxbi hþ1 hþ1 ðh þ 1Þ
By solving this inverse dispersion matrix, these solution will yield the asymptotic variance and co-variances of these ML ^ and ^ h; b a. By using Eq. (23), approximately estimators for ^ 100(1 a)% confidence intervals for h,b and a can be determined as ^ h Za2
qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ^ Za V b11 b b22 ^ b33 ; a a Z V V 2 2
902
S.K. Ashour and M.A. Eltehiwy
where Za2 is the upper ath percentile of the standard normal distribution. Order statistics Suppose X1, X2, . . . , Xn is a random sample from Eq. (8). Let X(1) < X(2) < < X(n) denote the corresponding order statistics. It is well known that the probability density function and the cumulative distribution function of the kth order statistics, say Y = X(k), are given by
n n X nJ X X nJ n J n nJ I F ðyÞf1 FðyÞg ¼ ð1Þ FJþI ðyÞ; I J J¼k J J¼k I¼0
respectively, for k = 1, 2, . . . , n. It follows from Eqs. (7) and (8) that aðkþjÞ1 b nk nk ah2 bn!yb1 ð1 þ yb Þehy X hyb b ð1Þj 1 1 þ ehy ; h þ 1ðk 1Þ!ðn kÞ! j¼0 hþ1 j
and
with respect to the unknown parameters. Therefore, in the case of EPL distribution the least square estimators of h, b and a, ^LSE and ^ say ^ hLSE ; b aLSE , respectively, can by obtained by minimizing the following equation (" ! #a )2 n X hxbðiÞ hxb i ðiÞ Qðh; b; aÞ ¼ 1 1þ ; ð25Þ e hþ1 nþ1 i¼0
(" n X
hxbðiÞ
(" n X
hxbðiÞ
FY ðyÞ ¼
J¼k I¼0
J
I
aðJþIÞ hxb hxb ð1Þ 1 1 þ e : hþ1 I
The qth moment of Y can be expressed as an! ðh þ 1Þðk 1Þ!ðn kÞ! 2 3 qþbðrþ1Þ qþbðrþ2Þ C C b b 6 7 þ qþrb 5; Aj;i;r 4 qþrb qþbðrþ1Þ qþbðrþ2Þ ði þ 1ÞCð b Þ h b 1 ði þ 1ÞCð b Þ h b
!
(" n X
1 1þ
"
1 1þ
hxbðiÞ
!
hxbðiÞ
#a
)" ! #a1 hxbðiÞ hxb i ðiÞ 1 1þ e 1 1þ e nþ1 hþ1 hþ1 i¼0 13 2 0 b h xðiÞ þ 1 hxb A5 ¼ 0; ð27Þ 4hxbðiÞ e ðiÞ ln xðiÞ @ hþ1
i¼0
n X nJ X n nJ
ð24Þ
)" ! #a1 hxbðiÞ hxb i ðiÞ 1 1þ e 1 1þ e nþ1 hþ1 hþ1 i¼0 n h io hxb ð26Þ e ðiÞ hð1 þ hÞ 1 þ h þ hxbðiÞ xbðiÞ ¼ 0;
and
fY ðyÞ ¼
i¼0
2 i ; nþ1
with respect to h, b and a. To minimize equation (Eq. (25)) with respect to h, b and a. We differentiate it with respect to these parameters, which leads to the following equations
n! fY ðyÞ ¼ Fk1 ðyÞf1 FðyÞgnk fðyÞ ðk 1Þ!ðn kÞ! nk X nk n! ¼ ð1Þj Fk1þj ðyÞfðyÞ; ðk 1Þ!ðn kÞ! j¼0 j
FY ðyÞ ¼
n X Qðh; b; aÞ ¼ F xðiÞ
hxbðiÞ
hxbðiÞ
!
#a
hxbðiÞ
e hþ1 ! #a
hþ1
hxbðiÞ
e
#a
) i nþ1 "
ln 1 1 þ
hxbðiÞ
!
hþ1
hxbðiÞ
e
# ¼ 0: ð28Þ
By solving this nonlinear system of Eqs. (26)–(28), this solution ^LSE and ^ will yield the LSE estimators ^ hLSE ; b aLSE .
EðYq Þ ¼
where Aj;i;r ¼ r h ð1Þiþj . hþ1
Pnk P1 P1 j¼0
i¼0
r¼0
nk j
aðk þ jÞ 1Þ i
i r
Least square estimation In this section, we provide the regression based method estimators of the unknown parameters, which was originally suggested by Swain et al. [13] to estimate the parameters of Beta distributions. The method can be described as follows: Suppose X1, X2, . . . , Xn be a random sample of EPLD(h, b, a) exponentiated power Lindley distribution with cdf F(x), and suppose that X(i), i = 1, 2, . . . , n denote the ordered sample. It is well known that
i E F xðiÞ ¼ E P X 6 xðiÞ ¼ : nþ1 (See, Johnson et al. 14). The least square estimators (LSES) are obtained by minimizing
Data analysis In this section we provide a data analysis in order to assess the goodness-of-fit of EPL model with respect to maximum flood levels data to see how the new model works in practice. The data have been obtained from Dumonceaux and Antle [15]. We fit the EPL distribution to the real data set and compare its fitting with some usual survival distributions. Namely, i. The modified Weibull (MW) distribution [16] with pdf given by b fðxÞ ¼ h þ abxb1 ehxax ; x > 0; h; b; a > 0; where a and b are the shape parameters and h is the scale parameter. ii. The exponentiated exponential (EE) distribution [17] with pdf given by b1 fðxÞ ¼ hbehx 1 ehx ; x > 0; h; b > 0; where h is the scale parameter and b is a shape parameter (see Table 1). iii. The Weibull (W) distribution with pdf given by b fðxÞ ¼ hbxb1 ehx ;
x > 0;
h; b > 0:
Exponentiated power Lindley distribution Since the power Lindley (PLD), generalized Lindley (GLD) and Lindley (LD) distributions are special cases of the exponentiated power Lindley distribution, we fit them to these data as well. The analysis of least square estimates for the unknown parameters in the seven fitted distributions by using the method of least squares, is defined. The LSE(s) of the unknown parameter(s), coefficient of determination (R2) and the corresponding Mean square error of the distributions mentioned before are given in Table 2. It is clear that the exponentiated power Lindley (EPLD) distribution provides better fit than the other distributions. Another check is to compare the respective coefficients of determination for these regression lines. We have supporting evidence that the coefficient of determination of (EPLD) is 0.975, which is higher than the coefficient of determination (R2) of (PLD), (GLD), (LD), (EE), (MW) and (WD) distributions. Hence the data point from the exponentiated power Lindley distribution (EPLD) has better relationship and hence this distribution is good model for life time data. As a second application, we analyze a real data set on the active repair times (h) for an airborne communication transceiver. The data are given in Table 3, and their source is Jorgensen [18]. In order to compare distributions we consider the ^ ^hÞ values, the Akaike information criteLOG ¼ log Lð^a; b; rion (AIC) and Bayesian information criterion (BIC), which are defined, respectively, by 2LOG + 2q and 2LOG + qlog(n), ^ ^hÞ are the MLEs vector, q is the number of paramwhere ð^a; b; eters estimated and n is the sample size. The best distribution corresponds to lower LOG, AIC and BIC values. Table 4 shows the values of the AIC, BIC and LOG, and also the Kolmogorov–Smirnov statistic with their p values. Table 5 shows the parameter MLEs according to each one of the seven fitted distributions. The values of AIC, BIC, LOG and K–S statistic with their p value in Table 4, indicate that the EPLD distribution is a strong competitor to other distributions commonly used in literature for fitting lifetime data, moreover being the best fitting considering AIC, BIC, LOG and K–S criterion. Simulation study We used a simulation study to investigate the performance of the accuracy of point and interval estimates of the EPL(a, b, h). The following steps are as follows: 1. Specify the values of the parameters a, band h; 2. Specify the sample size n; 3. Use Algorithm IV to generate a random sample with size n from EPL(a, b, h). a. Calculate the MLE of the three parameters and the inverse of the Fisher matrix. b. Calculate the squared deviation of the MLE from the exact value of each parameter. c. Calculate a 95% CI for each parameter.
Table 1 Maximum flood levels data from Dumonceaux and Antle [15]. 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.3235 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265
903 Table 2 Estimated parameters of EPLD, PLD, GLD, LD and WD distributions. Distribution
h
b
a
MSE
R2
EPLD PLD GLD LD EE MW WD
11.465 50.001 10.884 2.353 9.011 0.0365 49.025
0.774 4.69 – – 24.385 0.05 4.69
131.759 – 36.675 – – 0.219 –
0.00224 0.003 0.00244 0.036 0.00277 0.0034 0.003
0.975 0.964 0.970 0.551 0.967 0.963 0.964
Table 3
Active repair times (h).
0.50 1.00 1.50 3.00 5.40
0.60 1.00 1.50 3.30 7.00
0.60 1.00 2.00 4.00 7.50
0.70 1.00 2.00 4.00 8.80
0.70 1.10 2.20 4.50 9.00
0.70 1.30 2.50 4.70 10.20
0.80 1.50 2.70 5.00 22.00
0.80 1.50 3.00 5.40 24.50
4. Repeat steps 2–3, N times; 5. Calculate the mean square error (MSE), the average of the confidence interval widths, and the coverage probability for each parameter. The MSE associated with MLE of the parameter #, MSE#, is MSE# ¼
N 2 1X #^i # ; N i¼1
where #^i is the MLE of # using the ith sample, i = 1, 2, . . . , N, and # = a, b, h. Coverage probability is the proportion of the N simulated confidence intervals which include the true parameter #.
Table 4
Comparison criterion.
Model
AIC
BIC
LOG
K–S statistic
p-Value
EPLD PLD GLD LD EE MW WD
186.5721 195.8854 199.8218 198.5826 194.9158 195.4046 195.0227
191.6387 199.2631 203.1995 201.2715 198.2936 200.4713 198.4005
90.2861 95.9427 97.9109 98.7913 95.4579 94.7023 95.5114
0.0909 0.1596 0.1410 0.1907 0.1334 0.1631 0.1540
0.8627 0.4092 0.4362 0.3424 0.4896 0.4004 0.3782
Table 5
Parameters MLES.
Distribution
h
b
a
EPLD PLD GLD LD EE MW WD
3.5472 0.5867 0.3588 0.4242 0.2678 0.3037 0.2688
0.2901 0.7988 – – – 1.7269 0.9604
30.8299 – 0.7460 – 1.1137 0.0106 –
904
S.K. Ashour and M.A. Eltehiwy
Table 6
MSE, coverage probability, and average width.
a
b
h
n
MSEa
MSEb
MSEh
CPa
AWa
CPb
AWb
CPh
AWh
1.5
1
1
25 50 75 100
1.053 0.365 0.206 0.141
0.035 0.015 0.009 0.006
0.060 0.029 0.016 0.012
0.957 0.954 0.952 0.955
4.126 1.430 0.808 0.552
0.955 0.953 0.949 0.951
0.669 0.458 0.369 0.318
0.949 0.954 0.956 0.949
0.886 0.609 0.493 0.425
1
2
0.1
25 50 75 100
0.285 0.092 0.056 0.040
0.058 0.024 0.015 0.012
0.017 0.008 0.005 0.004
0.962 0.956 0.955 0.951
1.118 0.362 0.218 0.156
0.953 0.953 0.955 0.955
0.866 0.594 0.481 0.414
0.926 0.934 0.940 0.945
0.481 0.339 0.277 0.239
1
0.6
2
25 50 75 100
0.187 0.062 0.038 0.026
0.009 0.004 0.002 0.001
0.063 0.025 0.015 0.012
0.960 0.954 0.953 0.955
0.734 0.244 0.150 0.100
0.955 0.955 0.956 0.954
0.336 0.229 0.185 0.159
0.942 0.954 0.952 0.953
0.888 0.608 0.492 0.426
0.8
0.2
10
25 50 75 100
0.450 0.201 0.123 0.094
0.003 0.002 0.001 0.001
1.471 0.506 0.295 0.198
0.958 0.952 0.955 0.951
5.150 1.669 0.947 0.645
0.932 0.938 0.942 0.952
0.135 0.092 0.074 0.064
0.947 0.948 0.947 0.951
3.941 2.505 1.976 1.679
1
0.88
1.2
25 50 75 100
0.078 0.036 0.024 0.014
0.036 0.015 0.009 0.007
0.089 0.038 0.024 0.017
0.962 0.957 0.955 0.951
0.160 0.056 0.033 0.018
0.950 0.952 0.952 0.948
0.674 0.457 0.368 0.318
0.955 0.952 0.954 0.952
1.075 0.735 0.594 0.510
1
0.9
1.5
25 50 75 100
0.151 0.075 0.050 0.036
0.063 0.025 0.016 0.012
0.175 0.065 0.040 0.029
0.963 0.959 0.956 0.950
0.652 0.231 0.137 0.072
0.953 0.951 0.959 0.954
0.875 0.596 0.481 0.414
0.960 0.953 0.953 0.955
1.415 0.945 0.759 0.652
0.05
2
2
25 50 75 100
0.064 0.028 0.020 0.014
0.144 0.061 0.037 0.028
0.060 0.026 0.016 0.012
0.962 0.956 0.950 0.949
0.135 0.049 0.030 0.017
0.952 0.947 0.953 0.949
1.342 0.916 0.740 0.638
0.943 0.945 0.950 0.951
0.883 0.608 0.492 0.424
0.09
3
1
25 50 75 100
0.072 0.033 0.019 0.012
0.148 0.061 0.038 0.028
0.092 0.038 0.024 0.017
0.965 0.955 0.955 0.950
0.161 0.057 0.032 0.018
0.951 0.952 0.949 0.947
1.344 0.916 0.738 0.636
0.949 0.952 0.949 0.954
1.076 0.734 0.593 0.509
The simulation study is used when N = 10,000, the sample sizes are 25, 50, 75, 100, and the parameters values (a, b, h) = (1.5, 1, 1), (1, 2, 0.1), (1, 0.6, 2), (0.8, 0.2, 10), (1, 0.88, 1.2), (1, 0.9, 1.5), (0.05, 2, 2), (0.09, 3, 1). Some of the selected values of (a, b, h) give increasing, decreasing, increasing–decreasing–increasing, bath tub hazard shapes, respectively as shown in Fig. 2. Table 6 presents the MSE, Coverage probability (CP#), and average width (AW) of 95% confidence intervals of each parameter. As it was expected, this table shows that the MSEs of the estimates decrease as the sample size increases, that the coverage probabilities are very close to the nominal level of 95%, and that the average widths decrease as the sample size increases.
subject distribution can be used to model reliability data. We derived the maximum likelihood estimates of the parameters and their variance covariance matrix. A real data application of the EL distribution shows that it could provide a better fit than a set of usual statistical distributions considered in lifetime data analysis. Finally, we examined the accuracy of the maximum likelihood estimators of the EPL(a, b, h) parameters as well as the coverage probability and average width of the confidence intervals for the parameters using simulation. Conflict of Interest The authors have declared no conflict of interest. Compliance with Ethics Requirements
Conclusion In this study we have proposed a new family of distributions called exponentiated power Lindley distribution (EPLD). We get the probability density functions for generalized Lindley, Power Lindley, and Lindley distributions as special cases from ELPD. Some mathematical properties along with estimation issues are addressed. The hazard rate function behavior of the exponentiated power Lindley distribution shows that the
This article does not contain any studies with human or animal subjects. References [1] Ghitany ME, Al-Mutairi DK, Balakrishnan N, Al-Enezi LJ. Power Lindley distribution and associated inference. Comput Stat Data Anal 2013;64:20–33.
Exponentiated power Lindley distribution [2] Nadarajah S, Bakouch HS, Tahmasbi RA. Generalized Lindley distribution. Sankhya B 2011;73:331–59. [3] Lindley DV. Fiducial distributions and Bayes’ theorem. JR Stat Soc Ser A 1958;20:102–7. [4] Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its application. Math Comput Simulat 2008;78:493–506. [5] Bakouch HS, Al-Zahrani BM, Al-Shomrani AA, Marchi VAA, Louzada F. An extended Lindley distribution. J Korean Stat Soc 2012;41(1):75–85. [6] Ghitany ME, Al-qallaf F, Al-Mutairi DK, Hussain HA. A two parameter weighted Lindley distribution and its applications to survival data. Math Comput Simulat 2011;81(6):1190–201. [7] Merovci F, Elbatal I. Transmuted Lindley-geometric and its application. J Stat Appl 2014;3(1):77–91. [8] Asgharzadeh A, Bakouch SH, Nadarajah S, Esmaeili L. A new family of compound lifetime distributions. Kybernetika 2014;50(1):142–69. [9] Oluyede B, Yang T. A new class of generalized Lindley distributions with applications. J Stat Comput Simulat 2015;85(10):2072–100. [10] Shaked M, Shanthikumar JG. Stochastic orders and their applications. New York: Academic Press; 1994.
905 [11] Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DJ. On the Lambert Wfunction. Adv Comput Math 1996;5: 329–59. [12] Jodra´ J. Computer generation of random variables with Lindley or Poisson–Lindley distribution via the Lambert Wfunction. Math Comput Simulat 2010;81:851–9. [13] Swain J, Venkatraman S, Wilson J. Least squares estimation of distribution function in Johnson’s translation system. J Stat Comput Simulat 1988;29:271–97. [14] Johnson NL, Kotz S, Balakrishnan N. Continuous univariate distribution, 2nd ed., vol. 2. New York: John Wiley; 1995. [15] Dumonceaux R, Antle C. Discrimination between the lognormal and the Weibull distributions. Technometrics 1973;15(4):923–6. [16] Sarhan AM, Zaindin M. Modified Weibull distribution. Appl Sci 2009;11:123–36. [17] Gupta RD, Kundu D. Generalized exponential distributions. Aust NZ J Stat 1999;41(2):, 173–188. [18] Jorgensen B. Statistical properties of the generalized inverse Gaussian distribution. New York: Springer-Verlag; 1982.
Cairo University
Journal of Advanced Research
ORIGINAL ARTICLE
Exponentiated power Lindley distribution Samir K. Ashour a b
a,*
, Mahmoud A. Eltehiwy
b
Institute of Statistical Studies & Research, Department of Mathematical Statistics, Cairo University, Egypt Faculty of Commerce, Department of Statistics, South Valley University, Egypt
A R T I C L E
I N F O
Article history: Received 15 July 2014 Received in revised form 11 August 2014 Accepted 17 August 2014 Available online 24 August 2014 Keywords: Lambert function Least square estimation Maximum likelihood estimation Order statistics Power Lindley distribution
A B S T R A C T A new generalization of the Lindley distribution is recently proposed by Ghitany et al. [1], called as the power Lindley distribution. Another generalization of the Lindley distribution was introduced by Nadarajah et al. [2], named as the generalized Lindley distribution. This paper proposes a more generalization of the Lindley distribution which generalizes the two. We refer to this new generalization as the exponentiated power Lindley distribution. The new distribution is important since it contains as special sub-models some widely well-known distributions in addition to the above two models, such as the Lindley distribution among many others. It also provides more flexibility to analyze complex real data sets. We study some statistical properties for the new distribution. We discuss maximum likelihood estimation of the distribution parameters. Least square estimation is used to evaluate the parameters. Three algorithms are proposed for generating random data from the proposed distribution. An application of the model to a real data set is analyzed using the new distribution, which shows that the exponentiated power Lindley distribution can be used quite effectively in analyzing real lifetime data. ª 2014 Production and hosting by Elsevier B.V. on behalf of Cairo University.
Abbreviations: EPLD, Exponentiated power Lindley distribution; PLD, Power Lindley distribution; BGLD, Beta generalized Lindley distribution; GLD, Generalized Lindley distribution; LD, Lindley distribution; MW, Modified Weibull distribution; WD, Weibull distribution; EE, Exponentiated exponential distribution; Cdf, Cumulative distribution function; pdf, Probability density function; h(t), Hazard rate function; Q(p), Quantile function; E(Xr), The rth moment; MX(t), The moment generating function; MLE, Maximum likelihood estimator; LSE, Least square estimator; MSE, mean square error; L, Log-likelihood function; K–S, Kolmogorov– Smirnov test; AIC, Akaike information criterion; BIC, Bayesian information criterion. * Corresponding author. E-mail address: [email protected] (S.K. Ashour). Peer review under responsibility of Cairo University.
Production and hosting by Elsevier
Introduction Lindley [3], introduced a one-parameter distribution, known as Lindley distribution, given by its probability density function gðt; hÞ ¼
h2 ð1 þ tÞeht ; 1þh
t > 0;
h > 0:
ð1Þ
It can be seen that this distribution is a mixture of exponential (h) and gamma (2, h) distributions. Its cumulative distribution function has been obtained as GðtÞ ¼ 1
h þ 1 þ ht ht e ; hþ1
t > 0;
h > 0:
ð2Þ
Ghitany et al. [4] have discussed various properties of this distribution and showed that in many ways that the pdf given by (1) provides a better model for some applications than the
2090-1232 ª 2014 Production and hosting by Elsevier B.V. on behalf of Cairo University. http://dx.doi.org/10.1016/j.jare.2014.08.005
896
S.K. Ashour and M.A. Eltehiwy
exponential distribution. Bakouch et al. [5] obtained an extended Lindley distribution and discussed its various properties and applications. Ghitany et al. [6] developed a two-parameter weighted Lindley distribution and discussed its applications to survival data. Nadarajah et al. [2] obtained a generalized Lindley distribution and discussed its various properties and applications. Merovci and Elbatal [7] use the quadratic rank transmutation map in order to generate a flexible family of probability distributions taking Lindley-geometric distribution as the base value distribution by introducing a new parameter that would offer more distributional flexibility and called it transmuted Lindley-geometric distribution. Asgharzadeh et al. [8] introduced a general family of continuous lifetime distributions by compounding any continuous distribution and the Poisson–Lindley distribution. Oluyede and Yang [9] proposed a new four-parameter class of generalized Lindley (GLD) distribution called the beta-generalized Lindley distribution (BGLD). This class of distributions contains the betaLindley, GLD and Lindley distributions as special cases. A two parameter power Lindley distribution (PLD), of which the Lindley distribution ‘Eq. (1)’ is a particular case, has been suggested by Ghitany et al. [1]. They introduced a new extension of the Lindley distribution by considering the power transformation of the r.v. Y = T1/b. The pdf of the Y is readily obtained to be power Lindley distribution with parameters b and h and is defined by its probability density function pdf gðy; hÞ ¼
h2 bð1 þ yb Þ b1 hyb y e ; hþ1
y > 0;
h;
b > 0;
ð3Þ
It can easily be seen that at b = 1 , Eq. (3) reduces to the Lindley distribution. From Eq. (2), we see that the power Lindley distribution is a two-component mixture of Weibull distribution (with shape b and scale h), and a generalized gamma distribution (with shape parameters 2 b and scale h), with mixing h proportion p ¼ 1þh . gðy; a; hÞ ¼ pf1 ðyÞ þ ð1 pÞf2 ðyÞ; where p ¼
ð4Þ
h , 1þh b
f1 ðyÞ ¼ hbyb1 ehy ;
y > 0;
h;
b > 0;
ð5Þ
and b
f2 ðyÞ ¼ h2 by2b1 ehy ;
y > 0;
h;
b > 0:
ð6Þ
In this paper, we introduce a new distribution with three parameters, referred to as the exponentiated power Lindley (EPLD) distribution, with the hope that it will attract many applications in different disciplines such as survival analysis, reliability, biology and others. One of the main goals to introduce this new distribution was that it involves three distributions as sub-models. Generally, the EPLD distribution generalizes the generalized Lindley (GLD) [2], power Lindley (PLD) [1] and Lindley (LD) [3] distributions. The third parameter indexed to this distribution makes it more flexible to describe different types of real data than its sub-models. The EPLD distribution, due to its flexibility in accommodating different forms of the hazard function, seems to be a suitable distribution that can be used in a variety of problems in fitting survival data. The EPLD distribution is not only convenient for fitting comfortable bathtub-shaped failure rate data but
also suitable for testing goodness-of-fit of some special submodels such as the PLD, GLD and LD distributions. The new extension of the power Lindley distribution is most conveniently specified in terms of the cumulative distribution function: a hxb b ehx ; FðxÞ ¼ 1 1 þ ð7Þ hþ1 for x > 0, h, b, a > 0 and the corresponding probability density function (pdf) is given by a1 ah2 bxb1 hxb b b 1 þ xb ehx 1 1 þ ehx fðxÞ ¼ ; ð8Þ hþ1 hþ1 The corresponding hazard rate function is ah2 b bxb1 1 þ xb ehx ðh þ 1Þ a1 hxb b 1 ehx 1 1þ fSðxÞg ; hþ1
hðxÞ ¼
ð9Þ
where a hxb b ehx : SðxÞ ¼ 1 1 1 þ hþ1
ð10Þ
Note that Eq. (8) has closed form survival functions and hazard rate functions. For (b = 1) , (a = 1) and (a = b = 1) we have the pdfs of generalized Lindley distribution , power Lindley and Lindley distributions respectively. As we shall see later, Eq. (8) has the attractive feature of allowing for monotonically decreasing; monotonically increasing and bath tub shaped hazard rate functions while not allowing for constant hazard rate functions. Another motivation for the new distribution in (Eq. (7)) can be described as follows. Consider the two parameters power Lindley distribution [1] specified by the cumulative distribution function: hxb b ehx ; ð11Þ FpL ðxÞ ¼ 1 1 þ hþ1 for x > 0 and h, b > 0. Suppose X1, X2 , . . . , Xa are independent random variables distributed according to (Eq. (11)) and represent the failure times of the components of a series system, assumed to be independent. Then the probability that the system will fail before time x is given by PrðmaxðX1 ;X2 ;. .. ;Xa Þ 6 xÞ ¼ PrðX1 6 xÞPrðX2 6 xÞ . .. PrðXa 6 xÞ ¼ FpL ðxÞFpL ðxÞ a hxb b . .. FpL ðxÞ ¼ 1 1 þ ehx : hþ1
So, Eq. (7) gives the distribution of the failure of a series system with independent components. From Eqs. (4)–(6), it follows that cumulative distribution function, Eq. (7), can be represented as 1 X a ai ai i FðxÞ ¼ ðxÞFGG2 ðxÞ; p ð1 pÞi FW i i¼0 b
where FW ðxÞ ¼ 1 eðhx Þ is the cumulative distribution function of Weibull random variable with shape b and scale h
Exponentiated power Lindley distribution
897
b
w(y) = [F(y)]a1 . The weight function is an increasing or decreasing if a > 1 or a < 1 respectively. Therefore, if Y is decreasing pdf then for a > 1, The pdf of X also is a decreasing pdf. If Y has a unimodal pdf, then Mode(X) > ( ( ( 0; a 6 1 ; II. 12 < b < 1; h P g1 ðbÞ; a 6 1 , pffiffiffiffiffiffiffiffiffiffi 12 bð1bÞ , where g1 ðbÞ ¼ b III. {b = 1, h P 1, a = 1}; IV. {b P 1, h > 0, a < 1}. b. Unimodal if I. {b = 1, h > 0, a > 1}; II. {b = 1, 0 < h < 1, a = 1}; III. {b > 1, h > 0, a P 1}. c. Decreasing–increasing–decreasing if 12 < b < 1; 0 < h < g1 ðbÞ; a ¼ 1Þ. Fig. 1 illustrates some of the possible shapes of the density function of exponentiated power Lindley distribution for selected values of the parameters (h, b, a). The behavior of h(x) at x = 0 and x = 1, respectively, are given by
Shapes In this section, we discuss the shape characteristics of the pdf f(x) in Eq. (8) of the EPL(h, b, a) distribution. The behavior of f(x) at x = 0 and x = 1, respectively, are given by 8 > < 1; if b < 1 or a < 1; h2 fð0Þ ¼ hþ1 ; if b ¼ 1 and a ¼ 1 ; fð1Þ ¼ 0: > : 0; if b > 1 or a > 1;
hð0Þ ¼
8 > < 1;
if b < 1 or a 6 1;
h2 ; > hþ1
if b ¼ 1 and a ¼ 1; : 0; if b > 1 or a P 1; 8 if b < 1 or a 6 1; > < 0; h2 ; if b ¼ 1 and a ¼ 1; hð1Þ ¼ hþ1 > : 1; if b > 1 or a P 1;
If X has the pdf (8), the pdf of X is a weighted version of the pdf of Y in Eq. (3) and the weight function in this case is 1.5
f ( x , 1 , 0.2 , .9) f ( x , 2 , 1 , 1) f ( x , 3 , 2 , .5)
0.8
f ( x , 1 , 1 , 2)
1
0.6
f ( x , .6 , 1 , 1)
f ( x , 1.5 , 2 , 1.5) 0.4
0.5
0.2 0
0
1
2
x
3
0
4
0
2
4
x
6
8
0.12 f ( x , .3 , .85 , 1) f ( x , .35 , .9 , 1)
0.1 0.08 0.06 0
Fig. 1
2
4
x
6
8
Probability density function of the EPLD(h, b, a) for selected values of the parameters.
898
S.K. Ashour and M.A. Eltehiwy 2
4
1.5
h ( x , 1 , 1 , 1.5) h ( x , .1 , 2 , 1)
3
h ( x , 2 , .6 , 1)
1
h ( x , 10 , .2 , .8)
0.5 0
2 1
0
2
4
x
6
8
0
10
0
2
0
1
4
6
8
2
3
4
x
10 8
1
h ( x , 1.2 , .88 , 1) h ( x , 1.5 , .9 , 1)
h ( x , 2 , 2 , .05) 6
0.9
h ( x , 1 , 3 , .09) 4
0.8 0.7 0
Fig. 2
2 2
4
x
6
8
0
10
x
Hazard rate function of the EPLD(h, b, a) for selected values of the parameters.
Ghitany et al .[1] discussed and proved the cases in which the hazard rate function of the power Lindley distribution is decreasing, increasing and decreasing–increasing–decreasing. It follows that the HRF h(x) in Eq. (9) of the EPLD distribution is (a) Decreasing if I. 0 < b 6 12 ; h > 0; a 6 1 ; ð2b1Þ2 II. 12 < b < 1; h P g2 ðbÞ; a 6 1 , where g2 ðbÞ ¼ 4bð1bÞ ; (b) Increasing if {b P 1, h > 0, a P 1}. (c) Decreasing–increasing–decreasing if 12 < b < 1; 0 < h < g2 ðbÞ; a ¼ 1g. Fig. 2 shows the HRF h(x) of the exponentiated power Lindley distribution for some choices of values of the parameters (h, b, a). Stochastic orders
Theorem 1. Let X1 EPLD(h1, b, a1) and X2 EPLD (h2, b, a2). If h1 = h2, and a1 > a2 (or if a1 = a2 and h1 P h2), then X1 6 LrX2 and hence X1 6 hrX2 and X1 6 stX2. Proof f1 ðxÞ a1 h21 ðh2 þ 1Þb1 xb1 1 b b ¼ ð1 þ xb1 Þeðh2 x 2 h1 x 1 Þ f2 ðxÞ a2 h22 ðh1 þ 1Þb2 xb2 1 a1 1 h1 xb1 b eh1 x 1 1 1þ ðh1 þ 1Þ 1a2 h2 xb2 b eh2 x 2 1 1þ ; ðh2 þ 1Þ thus @ f ðxÞ ¼ ln 1 @x f2 ðxÞ
Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behavior. Suppose Xi is distributed according to Eqs. (7) and (8) with common shape parameter b and parameters hi and ai for i = 1, 2. Let Fi denote the cumulative distribution of Xi and let fi denote the probability density function of Xi. A random variable X1 is said to be smaller than a random variable X2 in the I. Stochastic order ðX 1 6st X 2 Þ if F1(x) P F2(x) for all x. II. Hazard rate order ðX 1 6hr X 2 Þ if h1(x) P h2(x) for all x. III. Likelihood ratio order ðX 1 6Lr X 2 Þ if ff 12 ðxÞ decreases in x. ðxÞ The following results due to Shaked and Shanthikumar [10] are well known for establishing stochastic ordering of distributions X1 6Lr X2 ) X1 6hr X2 ) X1 6st X2 : The EPLD is ordered with respect to the strongest ‘‘likelihood ratio’’ ordering as shown in the following theorem:
b1 h1 xb1 1 b2 h2 xb2 1 b1 b2 þ h1 xb1 h2 xb2 x b þb 1 b1 1 1 2 x ðb1 b2 Þ þ b1 x b2 xb2 1 þ ðxb1 þ 1Þðxb2 þ 1Þ b
ða1 1Þh21 b1 xb1 1 ð1 þ xb1 Þeh1 x 1 b b ðh1 þ 1Þ 1 1 þ hh11xþ11 eh1 x 1
þ
ða2 1Þh22 b2 xb2 1 ð1 þ xb2 Þeh2 x 2 : b b ðh2 þ 1Þ 1 1 þ hh22xþ12 eh2 x 2
b
@ Case (i) if h1 = h2, and a1 > a2, then @x ln ff12 ðxÞ < 0. This means ðxÞ
that X1 6Lr X2 and hence X1 6hr X2 and X1 6st X2. @ Case (ii) if h1 P h2, and a1 = a2 = a, then @x ln ff1 ðxÞ ðxÞ < 0. 2
This means that X1 6Lr X2 and hence X1 6hr X2 and X1 6st X2.
h
Moments Theorem 2. The rth moments E(Xr) of a exponentiated power Lindley random variable X, is given by
Exponentiated power Lindley distribution
899
a EðXr Þ ¼ EðXr Þ ¼ Ci;k hþ1 2 3 C rþbðkþ1Þ C rþbðkþ2Þ b b 4 5; þ rþkb rþbðkþ1Þ rþbðkþ2Þ rþkb hð b 1Þ :ði þ 1ÞCð b Þ hð b Þ :ði þ 1ÞCð b Þ where Ci;k ¼
P1 P1 i¼0
k¼0
a1 i
Theorem 3. Let X have an exponentiated power Lindley distribution. Then the moment generating function of X, MX(t), is 3 2 C jþbðkþ1Þ C jþbðkþ2Þ b b a 7 6 MX ðtÞ ¼ Ci;k;j 4 jþkb þ jþkb 5; jþbðkþ1Þ jþbðkþ2Þ hþ1 ði þ 1ÞCð b Þ h b 1 ði þ 1ÞCð b Þ hb
k i h ð1Þi hþ1 k
Proof Z
1
xr fðxÞdx;
MX ðtÞ ¼
0
k¼0
j¼0
k i h tj ð1Þi hþ1 . j! k
a1 i
Z
Z
1
etx fðxÞdx; 0
a1 ah2 bxrþb1 hxb b b 1 þ xb ehx 1 1 þ ehx dx hþ1 hþ1 0 Z 1 2 rþb1 a1 ah bx hxb b b ¼ ehx ehx 1 1 þ dx hþ1 hþ1 0 a1 Z 1 2 rþ2b1 ah bx hxb b b þ ehx ehx 1 1 þ dx: ð12Þ hþ1 hþ1 0
EðXr Þ ¼
1
MX ðtÞ ¼
Z
1 0
¼
Z
1 0
þ
Z
a1 ah2 bxb1 hxb b b 1 þ xb etx ehx 1 1 þ ehx dx hþ1 hþ1 a1 ah2 bxb1 tx hxb hxb b ehx e e 1 1þ dx hþ1 hþ1
1
0
Using the following binomial h ia1 b hxb ehx given by 1 1 þ hþ1
series
expansion
of
a1 X i 1 a1 hxb hxb b b 1 1þ ¼ eihx ; ehx ð1Þi 1 þ hþ1 h þ 1 i i¼0
Eq. (12) takes the following form k Z 1 2 b1þrþkb 1 X i h ah bx b ehx ðiþ1Þ dx h þ 1 h þ 1 i k 0 i¼0 k¼0 k Z 1 2 2b1þrþkb 1 1 X X a1 i h ah bx b ð1Þi ehx ðiþ1Þ dx þ hþ1 hþ1 i 0 i¼0 k¼0 k
EðXr Þ ¼
1 X a1
Let t = x and Ci;k ¼
P1
i¼0
k P a1 i h ð1Þi 1 . k¼0 k hþ1 i
EðX Þ ¼ Ci;k
Z
1
0
rþkb
ah2 t b htðiþ1Þ e dt þ Ci;k hþ1
Z 0
1
of
Eq. (13) takes the following form MX ðtÞ ¼
1 a1 X
þ
! i
ð1Þ
i
i¼0
1 a1 X
1 X
i
k¼0
k
! ð1Þi
i
i¼0
1 X k¼0
!
h hþ1
! i k
k Z
1 0
h hþ1
k Z
1 0
ah2 bxb1þkb tx hxb ðiþ1Þ e e dx hþ1 ah2 bx2b1þkb tx hxb ðiþ1Þ e e dx: hþ1
Using the following expansion of etx given by 1 j j X tx j¼0
j!
:
Eq. (13) can be rewritten as follows:
rþbðkþ1Þ
ah2 t b hþ1
expansion
h ii hxb and using the following binomial series expansion of 1 þ hþ1 given by i X k 1 i hxb h 1þ ¼ xkb : hþ1 hþ1 k k¼0
etx ¼
Eq. (12) can be rewritten as follows:
series
a1 X i 1 a1 hxb hxb b b i ehx ð1Þ 1 þ 1 1þ ¼ eihx ; hþ1 hþ1 i i¼0
ð1Þi
b
a1 ah2 bx2b1 tx hxb hxb b ehx e e 1 1þ dx: ð13Þ hþ1 hþ1
Using the following binomial h ia1 b hxb ehx given by 1 1 þ hþ1
h ii hxb and using the following binomial series expansion of 1 þ hþ1 given by i X k 1 i hxb h 1þ ¼ xkb ; h þ 1 hþ1 k k¼0
r
i¼0
Proof
EðXr Þ ¼
P1 P1 P1
where Ci;k;j ¼
htðiþ1Þ
e
dt; MX ðtÞ ¼
1 a1 X
3 C rþbðkþ1Þ C rþbðkþ2Þ b b a 6 7 EðXr Þ ¼ Ci;k 4 rþkb þ rþkb 5: rþbðkþ1Þ rþbðkþ2Þ hþ1 h b 1 ði þ 1ÞCð b Þ h b ; ði þ 1ÞCð b Þ
þ
1 X
a1
b
1 X
i
k¼0
k
!
i
i¼0
Let t = x k h tj . j! hþ1
i
ð1Þ
i
i¼0
C rþbðkþ1Þ b a a r EðX Þ ¼ Ci;k rþkb þ Ci;k rþbðkþ1Þ Cð b Þ 1Þ hþ1 h þ 1 ð b ði þ 1Þ h C rþbðkþ2Þ b rþkb ; rþbðkþ2Þ ð Þ b h ði þ 1ÞCð b Þ
!
ð1Þi
!
1 X
i
k¼0
k
h hþ1
!
and Ci;k;j ¼
k X 1 j Z 1 t ah2 bxb1þkbþj hxb ðiþ1Þ e dx j! hþ1 0 j¼0
h hþ1
k X 1 j Z 1 t ah2 bx2b1þkbþj hxb ðiþ1Þ e dx: j! hþ1 0 j¼0
P1 P1 P1 i¼0
k¼0
j¼0
a1 i
i ð1Þi k
Eq. (13) can be rewritten as follows:
2
MX ðtÞ ¼ Ci;k;j
Z
1 0
kbþj
ah2 t b htðiþ1Þ e dx þ Ci;k hþ1
Z
1 0
bðkþ1Þþj
ah2 t b hþ1
ehtðiþ1Þ dx;
900
S.K. Ashour and M.A. Eltehiwy
jþbðkþ1Þ
Algorithm I. (mixture form of the Lindley distribution)
C b a MX ðtÞ ¼ Ci;k;j jþkb jþbðkþ1Þ 1 hþ1 ði þ 1ÞCð b Þ hb
þ
jþbðkþ2Þ b
1. 2. 3. 4.
C a Ci;k;j jþkb ; jþbðkþ2Þ hþ1 ði þ 1ÞCð b Þ hb
2 3 C jþbðkþ1Þ C jþbðkþ2Þ b b a 6 7 MX ðtÞ ¼ Ci;k;j 4 jþkb þ jþkb 5: jþbðkþ1Þ jþbðkþ2Þ hþ1 ði þ 1ÞCð b Þ h b 1 ði þ 1ÞCð b Þ hb
Quantile function Let X denotes a random variable with the probability density function (Eq. (8)). The quantile function, say Q(p), defined by F(Q(p)) = p is the root of the equation # n o h½QðpÞb exp h½QðpÞb ¼ 1 p1=a ; 1þ 1þh
Generate Ui Unifrom(0, 1), i = 1, . . . , n; Generate Vi Exponential(h), i = 1, . . . n; Generate Gi Gamma(2, h), i = 1, . . . n; 1=a 1=b h , then set Xi ¼ Vi , otherwise, If Ui 6 p ¼ 1þh 1=b set Xi ¼ Gi ; i ¼ 1; . . . n.
Algorithm II. (mixture form of the power Lindley distribution) 1. 2. 3. 4.
Generate Ui Unifrom(0, 1), i = 1, . . . , n; Generate Yi Weibul(h, b), i = 1, . . . n; Generate Si GG(2, h, b), i = 1, . . . n; 1=a h If Ui 6 p ¼ 1þh , then set Xi = Yi, otherwise, set Xi = Si,i = 1, . . . n.
"
ð14Þ
for 0 < p < 1. Substituting Z(p) = 1 h h[Q(p)]b, one can rewrite Eq. (14) as
Algorithm III. (inverse cdf) 1. Generate Ui Unifrom(0, 1), i = 1, . . . , n; 2. Set
ZðpÞexpfZðpÞg ¼ ð1 þ hÞexpð1 hÞð1 p1=a Þ;
Xi ¼
for 0 < p < 1. So, the solution for Z(p) is ZðpÞ ¼ Wðð1 þ hÞexpð1 hÞð1 p1=a ÞÞ;
i1=b 1 1 h 1 W ð1 þ hÞexpð1 hÞ 1 U1=a ; i h h
ð15Þ where W(.) denotes the Lambert W function.
for 0 < p < 1, where W(.) is the Lambert W function, see Corless et al. [11] for detailed properties. Inverting Eq. (15), one obtains 1=b 1 1 1=a QðpÞ ¼ 1 Wðð1 þ hÞexpð1 hÞð1 p ÞÞ ; h h ð16Þ for 0 < p < 1. The particular case of Eq. (16) for (a = b = 1) has been derived recently by Jodra´ [12].
Algorithm IV 1. Generate Ui Unifrom(0, 1), i = 1, . . . , n; 2. Solve numerically the following equation in v 2 (0, 1):
vi ½h þ 1 ln vi ¼ 0: ðh þ 1Þ 1 U1=a i 3. Set Xi ¼
Generation algorithms Here, we consider simulating values of a random variable X with the probability density function in Eq. (8). Let U denote a uniform random variable on the interval (0, 1). One way to simulate values of X is to set hxb exp hxb ¼ 1 U1=a ; 1þ 1þh and solve for X, i.e. use the inversion method. Using Eq. (16), we obtain X as X¼
1=b 1 1 1 W ð1 þ hÞexpð1 hÞ 1 U1=a ; h h
where W[.] denotes the Lambert W function. We propose three different algorithms for generating random data from the exponentiated power Lindley distribution:
ln vi 1=b h
.
Maximum likelihood estimation of parameters Let x1, . . . , xn be a random sample of size n from EPLD. Then, the log-likelihood function is given by
Lða; b; hÞ ¼
n X
ln fðxi Þ;
i¼1
¼ n½ln ðaÞ þ ln ðbÞ þ 2 lnðhÞ lnðh þ 1Þ n n X X ln 1 þ xbi þ ðb 1Þ ln ðxi Þ þ i¼1
i¼1
n n X X ln Ai ðb; hÞ: h xbi þ ða 1Þ i¼1
i¼1
h i b hxb where Ai ðb; hÞ ¼ 1 1 þ hþ1i ehxi ; i ¼ 1; . . . ; n.
ð17Þ
Exponentiated power Lindley distribution
901
^ ^ The MLEs ^h; b; a of h, b, a are then the solutions of the following non-linear equations: n @ nðh þ 2Þ X Lða;b;hÞ ¼ xb @h hðh þ 1Þ i¼1 i " # n ða 1Þ X Ai;h ðb;hÞ þ ¼ 0; ð1 þ hÞ2 i¼1 Ai ðb;hÞ
ð18Þ
n n @ n X xbi ln ðxi Þ X Lða; b; hÞ ¼ þ þ ln ðxi Þ b @b b i¼1 x þ 1 i¼1 n n X X Ai;b ðb; hÞ ¼ 0; h xbi :ln ðxi Þ þ ða 1Þ Ai ðb; hÞ i¼1 i¼1
ð19Þ
n @ n X Lða; b; hÞ ¼ þ ln Ai ðb; hÞ ¼ 0; @a a i¼1
ð20Þ
where
2 n n n X @2L X xbi ln ðxi Þ2 n X x2b 2 b i ln ðxi Þ ¼ hx ln ð x Þ i i 2 @b2 b2 i¼1 xbi þ 1 xbi þ 1 i¼1 i¼1 h i
2 n ða 1Þ Ai ðb; hÞ:A 2 ðb; hÞ Ai;b ðb; hÞ X i;b þ ; ½Ai ðb; hÞ2 i¼1
" # @2L 2 1 ¼ n 2 @h2 h ðh þ 1Þ2 h i 2 n ða 1Þ Ai ðb; hÞA 2 ðb; hÞ ½Ai;h ðb; hÞ X i;h ; þ ½Ai ðb; hÞ2 i¼1
n X @2L Ai;h;a ðb; hÞ ¼ ; @h@a Ai ðb; hÞ i¼1
From (20) we can obtain the MLE of a as a function of (b, h), say ^aðb; hÞ, where n : ln Ai ðb; hÞ i¼1
@ 2 L n ¼ 2 ; @a2 a
n X @2L Ai;b;a ðb; hÞ ¼ ; @b@a Ai ðb; hÞ i¼1
b @Ai ðb; hÞ Ai;h ðb; hÞ ¼ ¼ ehxi hð1 þ hÞ 1 þ h þ hxbi xbi ; @h " # h xbi þ 1 @Ai ðb; hÞ b hxbi : Ai;b ðb; hÞ ¼ ¼ hxi e ln ðxi Þ @b hþ1
^ aðb; hÞ ¼ Pn
The second derivatives of L can be derived as follows:
ð21Þ
Putting ^aðb; hÞ in (17), we obtain n X gðb; hÞ ¼ L½^aðb; hÞ; b; h ¼ C n ln ½ ln Ai ðb;hÞ
n X @2L ¼ xbi ln ðxi Þ @h@b i¼1
n X ða 1Þ Ai ðb; hÞAi;h;b ðb; hÞ Ai;h ðb; hÞAi;b ðb; hÞ þ ; ½Ai ðb; hÞ2 i¼1
where
i¼1
n X
þ
b
Ai;h2 ðb; hÞ ¼
ln Ai ðb; hÞ þ n½ln ðbÞ þ 2 lnðhÞ lnðh þ 1Þ
i¼1 n X
n n X X ln 1 þ xbi þ ðb 1Þ ln ðxi Þ h xbi :
i¼1
i¼1
ð22Þ
i¼1
^MLE ; ^hMLE , can be obtained Therefore, the MLE of b, h, say b by maximizing (22) with respect to b and h. For the three parameters exponentiated power Lindley distribution EPLD(h, b, a), all the second order derivatives exist. Thus we have the inverse dispersion matrix is 20 1 0 13 0 1 ^ b11 V b12 V b13 h h V 7 6B C B b B^C b22 V b23 C A5; @ b A N4@ b A; @ V V 21 a
^ a 20
V1
V11 6B ¼ E4@ . . . V31
b31 V
... ... ...
b32 V
13 0 @2 L V13 2 C7 B @h . . . A5 ¼ E@ . . . @2 L V33 @h@a
...
@2 L @h@a
@2L @h@b @2L V23 ¼ @b@a V12 ¼
@2L ; @h@a @2L V33 ¼ 2 : @a V13 ¼
b hxi þ x2b i e
ðh þ 1Þ3
b Ai;b2 ðb; hÞ ¼ hxbi ln ðxi Þ2 ehxi
("
" b
Ai;b;a ðb; hÞ ¼ hxbi ln ðxi Þehxi " Ai;h;a ðb; hÞ ¼
... ...
@2 L @a2
1
"
2
!
ðh þ 1Þ2
!# hxbi 1þ ; hþ1
! # " #) h2 x2b 2hxbi 1 b i þ hxi 1 ; hþ1 hþ1
# hxbi þ h ; hþ1
! # hxbi 1 ; þ1 hþ1 ðh þ 1Þ2
b xbi ehxi
b
C . . . A:
Eq. (23) is the variance covariance matrix of the EPLD(h, b, a) @2L @h2 @2L V22 ¼ 2 @b
!
b33 V
ð23Þ
V11 ¼
2xbi ehxi
"
Ai;h;b ðb;hÞ ¼ xbi ehxi ln ðxi Þ
! !# hxbi hxb 1 hxbi þ h : þ 1 þ i 2 hxbi hþ1 hþ1 ðh þ 1Þ
By solving this inverse dispersion matrix, these solution will yield the asymptotic variance and co-variances of these ML ^ and ^ h; b a. By using Eq. (23), approximately estimators for ^ 100(1 a)% confidence intervals for h,b and a can be determined as ^ h Za2
qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ^ Za V b11 b b22 ^ b33 ; a a Z V V 2 2
902
S.K. Ashour and M.A. Eltehiwy
where Za2 is the upper ath percentile of the standard normal distribution. Order statistics Suppose X1, X2, . . . , Xn is a random sample from Eq. (8). Let X(1) < X(2) < < X(n) denote the corresponding order statistics. It is well known that the probability density function and the cumulative distribution function of the kth order statistics, say Y = X(k), are given by
n n X nJ X X nJ n J n nJ I F ðyÞf1 FðyÞg ¼ ð1Þ FJþI ðyÞ; I J J¼k J J¼k I¼0
respectively, for k = 1, 2, . . . , n. It follows from Eqs. (7) and (8) that aðkþjÞ1 b nk nk ah2 bn!yb1 ð1 þ yb Þehy X hyb b ð1Þj 1 1 þ ehy ; h þ 1ðk 1Þ!ðn kÞ! j¼0 hþ1 j
and
with respect to the unknown parameters. Therefore, in the case of EPL distribution the least square estimators of h, b and a, ^LSE and ^ say ^ hLSE ; b aLSE , respectively, can by obtained by minimizing the following equation (" ! #a )2 n X hxbðiÞ hxb i ðiÞ Qðh; b; aÞ ¼ 1 1þ ; ð25Þ e hþ1 nþ1 i¼0
(" n X
hxbðiÞ
(" n X
hxbðiÞ
FY ðyÞ ¼
J¼k I¼0
J
I
aðJþIÞ hxb hxb ð1Þ 1 1 þ e : hþ1 I
The qth moment of Y can be expressed as an! ðh þ 1Þðk 1Þ!ðn kÞ! 2 3 qþbðrþ1Þ qþbðrþ2Þ C C b b 6 7 þ qþrb 5; Aj;i;r 4 qþrb qþbðrþ1Þ qþbðrþ2Þ ði þ 1ÞCð b Þ h b 1 ði þ 1ÞCð b Þ h b
!
(" n X
1 1þ
"
1 1þ
hxbðiÞ
!
hxbðiÞ
#a
)" ! #a1 hxbðiÞ hxb i ðiÞ 1 1þ e 1 1þ e nþ1 hþ1 hþ1 i¼0 13 2 0 b h xðiÞ þ 1 hxb A5 ¼ 0; ð27Þ 4hxbðiÞ e ðiÞ ln xðiÞ @ hþ1
i¼0
n X nJ X n nJ
ð24Þ
)" ! #a1 hxbðiÞ hxb i ðiÞ 1 1þ e 1 1þ e nþ1 hþ1 hþ1 i¼0 n h io hxb ð26Þ e ðiÞ hð1 þ hÞ 1 þ h þ hxbðiÞ xbðiÞ ¼ 0;
and
fY ðyÞ ¼
i¼0
2 i ; nþ1
with respect to h, b and a. To minimize equation (Eq. (25)) with respect to h, b and a. We differentiate it with respect to these parameters, which leads to the following equations
n! fY ðyÞ ¼ Fk1 ðyÞf1 FðyÞgnk fðyÞ ðk 1Þ!ðn kÞ! nk X nk n! ¼ ð1Þj Fk1þj ðyÞfðyÞ; ðk 1Þ!ðn kÞ! j¼0 j
FY ðyÞ ¼
n X Qðh; b; aÞ ¼ F xðiÞ
hxbðiÞ
hxbðiÞ
!
#a
hxbðiÞ
e hþ1 ! #a
hþ1
hxbðiÞ
e
#a
) i nþ1 "
ln 1 1 þ
hxbðiÞ
!
hþ1
hxbðiÞ
e
# ¼ 0: ð28Þ
By solving this nonlinear system of Eqs. (26)–(28), this solution ^LSE and ^ will yield the LSE estimators ^ hLSE ; b aLSE .
EðYq Þ ¼
where Aj;i;r ¼ r h ð1Þiþj . hþ1
Pnk P1 P1 j¼0
i¼0
r¼0
nk j
aðk þ jÞ 1Þ i
i r
Least square estimation In this section, we provide the regression based method estimators of the unknown parameters, which was originally suggested by Swain et al. [13] to estimate the parameters of Beta distributions. The method can be described as follows: Suppose X1, X2, . . . , Xn be a random sample of EPLD(h, b, a) exponentiated power Lindley distribution with cdf F(x), and suppose that X(i), i = 1, 2, . . . , n denote the ordered sample. It is well known that
i E F xðiÞ ¼ E P X 6 xðiÞ ¼ : nþ1 (See, Johnson et al. 14). The least square estimators (LSES) are obtained by minimizing
Data analysis In this section we provide a data analysis in order to assess the goodness-of-fit of EPL model with respect to maximum flood levels data to see how the new model works in practice. The data have been obtained from Dumonceaux and Antle [15]. We fit the EPL distribution to the real data set and compare its fitting with some usual survival distributions. Namely, i. The modified Weibull (MW) distribution [16] with pdf given by b fðxÞ ¼ h þ abxb1 ehxax ; x > 0; h; b; a > 0; where a and b are the shape parameters and h is the scale parameter. ii. The exponentiated exponential (EE) distribution [17] with pdf given by b1 fðxÞ ¼ hbehx 1 ehx ; x > 0; h; b > 0; where h is the scale parameter and b is a shape parameter (see Table 1). iii. The Weibull (W) distribution with pdf given by b fðxÞ ¼ hbxb1 ehx ;
x > 0;
h; b > 0:
Exponentiated power Lindley distribution Since the power Lindley (PLD), generalized Lindley (GLD) and Lindley (LD) distributions are special cases of the exponentiated power Lindley distribution, we fit them to these data as well. The analysis of least square estimates for the unknown parameters in the seven fitted distributions by using the method of least squares, is defined. The LSE(s) of the unknown parameter(s), coefficient of determination (R2) and the corresponding Mean square error of the distributions mentioned before are given in Table 2. It is clear that the exponentiated power Lindley (EPLD) distribution provides better fit than the other distributions. Another check is to compare the respective coefficients of determination for these regression lines. We have supporting evidence that the coefficient of determination of (EPLD) is 0.975, which is higher than the coefficient of determination (R2) of (PLD), (GLD), (LD), (EE), (MW) and (WD) distributions. Hence the data point from the exponentiated power Lindley distribution (EPLD) has better relationship and hence this distribution is good model for life time data. As a second application, we analyze a real data set on the active repair times (h) for an airborne communication transceiver. The data are given in Table 3, and their source is Jorgensen [18]. In order to compare distributions we consider the ^ ^hÞ values, the Akaike information criteLOG ¼ log Lð^a; b; rion (AIC) and Bayesian information criterion (BIC), which are defined, respectively, by 2LOG + 2q and 2LOG + qlog(n), ^ ^hÞ are the MLEs vector, q is the number of paramwhere ð^a; b; eters estimated and n is the sample size. The best distribution corresponds to lower LOG, AIC and BIC values. Table 4 shows the values of the AIC, BIC and LOG, and also the Kolmogorov–Smirnov statistic with their p values. Table 5 shows the parameter MLEs according to each one of the seven fitted distributions. The values of AIC, BIC, LOG and K–S statistic with their p value in Table 4, indicate that the EPLD distribution is a strong competitor to other distributions commonly used in literature for fitting lifetime data, moreover being the best fitting considering AIC, BIC, LOG and K–S criterion. Simulation study We used a simulation study to investigate the performance of the accuracy of point and interval estimates of the EPL(a, b, h). The following steps are as follows: 1. Specify the values of the parameters a, band h; 2. Specify the sample size n; 3. Use Algorithm IV to generate a random sample with size n from EPL(a, b, h). a. Calculate the MLE of the three parameters and the inverse of the Fisher matrix. b. Calculate the squared deviation of the MLE from the exact value of each parameter. c. Calculate a 95% CI for each parameter.
Table 1 Maximum flood levels data from Dumonceaux and Antle [15]. 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.3235 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265
903 Table 2 Estimated parameters of EPLD, PLD, GLD, LD and WD distributions. Distribution
h
b
a
MSE
R2
EPLD PLD GLD LD EE MW WD
11.465 50.001 10.884 2.353 9.011 0.0365 49.025
0.774 4.69 – – 24.385 0.05 4.69
131.759 – 36.675 – – 0.219 –
0.00224 0.003 0.00244 0.036 0.00277 0.0034 0.003
0.975 0.964 0.970 0.551 0.967 0.963 0.964
Table 3
Active repair times (h).
0.50 1.00 1.50 3.00 5.40
0.60 1.00 1.50 3.30 7.00
0.60 1.00 2.00 4.00 7.50
0.70 1.00 2.00 4.00 8.80
0.70 1.10 2.20 4.50 9.00
0.70 1.30 2.50 4.70 10.20
0.80 1.50 2.70 5.00 22.00
0.80 1.50 3.00 5.40 24.50
4. Repeat steps 2–3, N times; 5. Calculate the mean square error (MSE), the average of the confidence interval widths, and the coverage probability for each parameter. The MSE associated with MLE of the parameter #, MSE#, is MSE# ¼
N 2 1X #^i # ; N i¼1
where #^i is the MLE of # using the ith sample, i = 1, 2, . . . , N, and # = a, b, h. Coverage probability is the proportion of the N simulated confidence intervals which include the true parameter #.
Table 4
Comparison criterion.
Model
AIC
BIC
LOG
K–S statistic
p-Value
EPLD PLD GLD LD EE MW WD
186.5721 195.8854 199.8218 198.5826 194.9158 195.4046 195.0227
191.6387 199.2631 203.1995 201.2715 198.2936 200.4713 198.4005
90.2861 95.9427 97.9109 98.7913 95.4579 94.7023 95.5114
0.0909 0.1596 0.1410 0.1907 0.1334 0.1631 0.1540
0.8627 0.4092 0.4362 0.3424 0.4896 0.4004 0.3782
Table 5
Parameters MLES.
Distribution
h
b
a
EPLD PLD GLD LD EE MW WD
3.5472 0.5867 0.3588 0.4242 0.2678 0.3037 0.2688
0.2901 0.7988 – – – 1.7269 0.9604
30.8299 – 0.7460 – 1.1137 0.0106 –
904
S.K. Ashour and M.A. Eltehiwy
Table 6
MSE, coverage probability, and average width.
a
b
h
n
MSEa
MSEb
MSEh
CPa
AWa
CPb
AWb
CPh
AWh
1.5
1
1
25 50 75 100
1.053 0.365 0.206 0.141
0.035 0.015 0.009 0.006
0.060 0.029 0.016 0.012
0.957 0.954 0.952 0.955
4.126 1.430 0.808 0.552
0.955 0.953 0.949 0.951
0.669 0.458 0.369 0.318
0.949 0.954 0.956 0.949
0.886 0.609 0.493 0.425
1
2
0.1
25 50 75 100
0.285 0.092 0.056 0.040
0.058 0.024 0.015 0.012
0.017 0.008 0.005 0.004
0.962 0.956 0.955 0.951
1.118 0.362 0.218 0.156
0.953 0.953 0.955 0.955
0.866 0.594 0.481 0.414
0.926 0.934 0.940 0.945
0.481 0.339 0.277 0.239
1
0.6
2
25 50 75 100
0.187 0.062 0.038 0.026
0.009 0.004 0.002 0.001
0.063 0.025 0.015 0.012
0.960 0.954 0.953 0.955
0.734 0.244 0.150 0.100
0.955 0.955 0.956 0.954
0.336 0.229 0.185 0.159
0.942 0.954 0.952 0.953
0.888 0.608 0.492 0.426
0.8
0.2
10
25 50 75 100
0.450 0.201 0.123 0.094
0.003 0.002 0.001 0.001
1.471 0.506 0.295 0.198
0.958 0.952 0.955 0.951
5.150 1.669 0.947 0.645
0.932 0.938 0.942 0.952
0.135 0.092 0.074 0.064
0.947 0.948 0.947 0.951
3.941 2.505 1.976 1.679
1
0.88
1.2
25 50 75 100
0.078 0.036 0.024 0.014
0.036 0.015 0.009 0.007
0.089 0.038 0.024 0.017
0.962 0.957 0.955 0.951
0.160 0.056 0.033 0.018
0.950 0.952 0.952 0.948
0.674 0.457 0.368 0.318
0.955 0.952 0.954 0.952
1.075 0.735 0.594 0.510
1
0.9
1.5
25 50 75 100
0.151 0.075 0.050 0.036
0.063 0.025 0.016 0.012
0.175 0.065 0.040 0.029
0.963 0.959 0.956 0.950
0.652 0.231 0.137 0.072
0.953 0.951 0.959 0.954
0.875 0.596 0.481 0.414
0.960 0.953 0.953 0.955
1.415 0.945 0.759 0.652
0.05
2
2
25 50 75 100
0.064 0.028 0.020 0.014
0.144 0.061 0.037 0.028
0.060 0.026 0.016 0.012
0.962 0.956 0.950 0.949
0.135 0.049 0.030 0.017
0.952 0.947 0.953 0.949
1.342 0.916 0.740 0.638
0.943 0.945 0.950 0.951
0.883 0.608 0.492 0.424
0.09
3
1
25 50 75 100
0.072 0.033 0.019 0.012
0.148 0.061 0.038 0.028
0.092 0.038 0.024 0.017
0.965 0.955 0.955 0.950
0.161 0.057 0.032 0.018
0.951 0.952 0.949 0.947
1.344 0.916 0.738 0.636
0.949 0.952 0.949 0.954
1.076 0.734 0.593 0.509
The simulation study is used when N = 10,000, the sample sizes are 25, 50, 75, 100, and the parameters values (a, b, h) = (1.5, 1, 1), (1, 2, 0.1), (1, 0.6, 2), (0.8, 0.2, 10), (1, 0.88, 1.2), (1, 0.9, 1.5), (0.05, 2, 2), (0.09, 3, 1). Some of the selected values of (a, b, h) give increasing, decreasing, increasing–decreasing–increasing, bath tub hazard shapes, respectively as shown in Fig. 2. Table 6 presents the MSE, Coverage probability (CP#), and average width (AW) of 95% confidence intervals of each parameter. As it was expected, this table shows that the MSEs of the estimates decrease as the sample size increases, that the coverage probabilities are very close to the nominal level of 95%, and that the average widths decrease as the sample size increases.
subject distribution can be used to model reliability data. We derived the maximum likelihood estimates of the parameters and their variance covariance matrix. A real data application of the EL distribution shows that it could provide a better fit than a set of usual statistical distributions considered in lifetime data analysis. Finally, we examined the accuracy of the maximum likelihood estimators of the EPL(a, b, h) parameters as well as the coverage probability and average width of the confidence intervals for the parameters using simulation. Conflict of Interest The authors have declared no conflict of interest. Compliance with Ethics Requirements
Conclusion In this study we have proposed a new family of distributions called exponentiated power Lindley distribution (EPLD). We get the probability density functions for generalized Lindley, Power Lindley, and Lindley distributions as special cases from ELPD. Some mathematical properties along with estimation issues are addressed. The hazard rate function behavior of the exponentiated power Lindley distribution shows that the
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